Here we provide the inner 300 × 300 of the full resolution and smoothed low resolution maps for both 610 MHz and 325 MHz. In each image, the dashed circle indicates the cluster scale θ500 = 3.10 from Hasselfield et al. (2013), centred on the SZ cluster peak, which is shown as a red or white X. The solid circle shows the 130 radius outside of which we removed all compact emission before further imaging in CASA, as described in§4.3.
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Figure 4.15: Inner 300 × 300 of the full-resolution (FR) 610 MHz map. The beam is 5.700 × 4.100 at p.a. 71.3◦, and the map noise isσ = 26µJy beam−1. The dashed black circle represents θ500 = 3.10, centred on the cluster SZ peak shown by the red X. The 130 radius is shown by the solid black circle.
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Figure 4.16: Inner 300 × 300 of the 610 MHz map. Greyscale is the low-resolution (LR), 10- smoothed image. Red contours are the high-resolution (HR) [6, 20, 80]×1σ contours where1σ
= 31µJy beam−1. The X and black solid and dashed circles are as in Figure 4.15. The LR beam is 79.600 ×76.800 at p.a. -86.9◦ and is shown by the blue ellipse in the lower left corner. The 1σ noise in the LR greyscale image is 0.36 mJy beam−1.
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Figure 4.17: Inner 300×300 of the full-resolution (FR) 325 MHz map. The beam is 9.700×7.900 at p.a. 74.1◦and the map noise isσ= 77µJy beam−1. The X and black solid and dashed circles are as in Figure 4.15.
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Figure 4.18: Inner 300 × 300 of the 325 MHz map. Greyscale is the low-resolution (LR), 10- smoothed image. Red contours are the high-resolution (HR) [6, 20, 80]×1σ contours where1σ
= 71µJy beam−1. The X and black solid and dashed circles are as in Figure 4.15. The LR beam is 79.400×73.100 at p.a. 56.7◦ and is shown by the blue ellipse in the lower left corner. The 1σ noise in the LR greyscale image is 1.18 mJy beam−1.
CLUSTER GRAVITATIONAL LENSING AND MASS MODELLING
General relativity predicts the local distortion of space-time around a large mass density. This leads to a geometric effect called gravitational lensing whereby light rays experience a local deformation along geodesics due to the distorted space-time. The first observational evidence of this phenomenon was in 1919 when Eddington measured the deviation of star positions by the Sun during an eclipse, however it wasn’t until 1979 when Walsh et al. (1979) observed the double-quasar Q0957+561 at a redshift of z=1.4 that the first lensed object was documented.
Gravitational lensing in clusters of galaxies was first observed in the mid- to late-80’s when giant gravitational arcs were detected in Abell 370, Abell 2218, and CL2244-02 (Soucail, 1987; Lynds and Petrosian, 1986). These arcs were hypothesised to be gravitationally lensed images of back- ground galaxies (Paczynski, 1987). This was confirmed by a redshift measurement ofz=0.724 for the arc in Abell 370 which lies at a redshift ofz=0.394 (Soucail et al., 1988). Following these discoveries, cluster gravitational lensing gained momentum as a field and is now used to trace the mass distribution of the lens (Mellier et al., 1993; Kneib et al., 1996) and to constrain cosmo- logical parameters (Jullo et al., 2010). In the following I will discuss the theory of gravitational
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lensing in the case of galaxy clusters and explain the lensing mass modelling techniques that I further used in the analysis of twoHubble Frontier Fields(HFF) clusters.
5.1 Gravitational lens equation
When a galaxy cluster of sufficient mass lies between an observer and a background source, the light from this source is deviated from the line-of-sight path causing distorted and sometimes multiple images of this source. Cluster gravitational lensing occurs in two regimes, categorised by the strength of the lensing effect on the background sources.Strong lensingeffects are visible in the high density region of the cluster core. Background sources that are approximately aligned along the line-of-sight with the cluster core will be strongly lensed. Background galaxies further away from this line-of-sight also experience the gravitational effect of the lens but the effect is weak, requiring a statistical detection. This is theweak lensingregime.
The gravitational lensing phenomenon links true positions of background sources in the source plane to observed source positions in the image plane. In order to describe the gravi- tational lensing formalism, several assumptions have to be made, the foremost being that the cosmological principle, namely that the Universe is isotropic and homogeneous, is true on large scales. Here large refers to the scalesLinvolved with the long-range gravitational force, i.e.
L∼ c
√Gρ¯∼2Gpc, (5.1)
wherecis the speed of light,Gis the gravitational constant, andρ¯is the mean density of the Uni- verse. The assumption of the cosmological principle imposes severe symmetries on the space- time metric. In highly concentrated regions, the metric will be locally perturbed, leading to the Schwarzschild metric (Weinberg, 1992) which describes a space-time near a point mass. In the stationary weak field limit Φ c2, and generalising for a continuous mass distribution, the metric becomes
ds2 =
1 + 2Φ c2
c2dt2−
1− 2Φ c2
dr2 (5.2)
whereΦis the 3D gravitational potential of the mass distribution involved.
Figure 5.1: Schematic of a single thin lens setup with observer O, lens L, background source S, and observed image I, showing the distances and angles relevant to the lens equation. Source:
adapted from Kneib and Natarajan (2011).
Consider the schematic in Figure 5.1 which depicts a simple single thin lens configuration in which an observer O views a source S through a lens L, observing an image I. Without an intervening lens, the observer would view the source at an angleθS. With the lens in place, the image of the source is instead observed at an angleθI, due to the deflection of the photon path coming from S, described by the deflection angleα, which itself depends on the local space-time deformation at θI. Since the local metric perturbation due to the lens, and thus the distortion angle, is minimal, the small-angle approximation oftanθ ≈θis valid. Using this in conjunction with Thales theorem on the triangle OSI, the geometric equation in the thin lens regime relating the position of the background galaxy in the source to image planes is
dθS =θI− DLS DOS
α(θI) =θI−εα(θI). (5.3)
The approximation of a thin lens is valid so long as the distances from the observer to the lens and to the source (DOLandDOSrespectively) are far greater than the physical extent of the lens.
This is always true in the case of galaxy and cluster lenses. The distance ratio ε = DLS/DOS is a function of the source redshift zS such that the higher the source redshift, the greater the
deflection and distortion. This ratio is used in the definition of the Einstein radius, defined as θE = εα when there is perfect alignment between the observer, lens, and source. The Einstein radius is used to quantify the strength of the lens. For a point mass the Einstein radius becomes
θE=
r4GM c2
ε
DOL, (5.4)
where the more massive the lens, the larger the geometric disruption will be.
Since photons follow null geodesics,ds2 = 0, one can determine the travel timetT for a given path length, which is a function of the deflection angle,α. Using Fermat’s principle, which states that light follows a path with a stationary travel time, we have dtT/dθI = 0and subsequently derive a formula for the deflection angle:
α(θI) = 2ε
c2∇θIφ2DN (θI) (5.5)
whereφ2DN (θI)is the Newtonian gravitational potential projected into the lens plane. Incorporat- ing this into equation 5.3 gives thelens equation in the thin lens approximation (for a detailed derivation see Schneider et al., 1992):
θS =θI− ∇θIϕ(θI), (5.6)
whereϕ(θI)is thelensing potentialand is defined as ϕ(θI) = 2ε
c2φ2DN (θI). (5.7)